Exact solution of a triaxial gyrostat with one rotor
The problem of the attitude dynamics of a triaxial gyrostat under no external torques and one constant internal rotor, is a three degrees-of-freedom system, although thanks to the existence of integrals of motion it can be reduced to only one degree-of-freedom problem. We introduce coordinates to represent the orbits of constant angular momentum as a flow on a sphere. This representation shows that the problem is equivalent to a quadratic Hamiltonian depending on two parameters. We find the exact solution of the orbits in terms of elliptic functions. By making use of properties of elliptic functions we find the solution at each region of the parametric partition from the solution of one region. We also prove that heteroclinic orbits are planar curves.
KeywordsAttitude motion Gyrostat Elliptic functions
- Byrd P.F., Friedman M.D.: Handbook of Elliptic Integrals for Engineers an Physicists. Springer Verlag, Berlin (1954)Google Scholar
- Hughes P.C.: Spacecraft Attitude Dynamics. Wiley, New York (1986)Google Scholar
- Lanchares, V.: Sistemas dinámicos bajo la acción del grupo SO(3): El caso de un Hamiltoniano cuadrático. Ph.D. dissertation, University of Zaragoza (in Spanish) (1993)Google Scholar
- Poinsot L.: Théorie nouvelle de la rotation des corps. J. Math. Pures Appl. 16, 289–336 (1851)Google Scholar